CEMC Banner

2019 Euclid Contest

Wednesday, April 3, 2019
(in North America and South America)

Thursday, April 4, 2019
(outside of North American and South America)

University of Waterloo Logo

©2019 University of Waterloo

Instructions

Time: \(2\frac{1}{2}\) hours

Number of Questions: 10
Each question is worth 10 marks.

Calculating devices are allowed, provided that they do not have any of the following features: (i) internet access, (ii) the ability to communicate with other devices, (iii) information previously stored by students (such as formulas, programs, notes, etc.), (iv) a computer algebra system, (v) dynamic geometry software.

Parts of each question can be of two types:

  1. SHORT ANSWER parts indicated by Lightbulb
  2. FULL SOLUTION parts indicated by Full Solution

WRITE ALL ANSWERS IN THE ANSWER BOOKLET PROVIDED.

Do not discuss the problems or solutions from this contest online for the next 48 hours.
The name, grade, school and location, and score range of some top-scoring students will be published on our website, cemc.uwaterloo.ca. In addition, the name, grade, school and location, and score of some top-scoring students may be shared with other mathematical organizations for other recognition opportunities.
NOTE:
  1. Please read the instructions for the contest.
  2. Write all answers in the answer booklet provided.
  3. For questions marked Lightbulb, place your answer in the appropriate box in the answer booklet and show your work.
  4. For questions marked Full solution, provide a well-organized solution in the answer booklet. Use mathematical statements and words to explain all of the steps of your solution. Work out some details in rough on a separate piece of paper before writing your finished solution.
  5. Diagrams are not drawn to scale. They are intended as aids only.
  6. While calculators may be used for numerical calculations, other mathematical steps must be shown and justified in your written solutions, and specific marks may be allocated for these steps. For example, while your calculator might be able to find the \(x\)-intercepts of the graph of an equation like \(y=x^{3} -x\), you should show the algebraic steps that you used to find these numbers, rather than simply writing these numbers down.

    1. LightbulbJoyce has two identical jars. The first jar is \(\frac{3}{4}\) full of water and contains 300 mL of water. The second jar is \(\frac{1}{4}\) full of water. How much water, in mL, does the second jar contain?

    2. LightbulbWhat integer \(a\) satisfies \(3<\dfrac{24}{a}<4\)?

    3. Full solutionIf \(\dfrac{1}{x^2} - \dfrac{1}{x} = 2\), determine all possible values of \(x\).

    1. LightbulbIn the diagram, two small circles of radius 1 are tangent to each other and to a larger circle of radius 2.

      The two smaller circles lie inside the larger circle. The larger circle is shaded. The two smaller circles are not.

      What is the area of the shaded region?

    2. LightbulbKari jogs at a constant speed of 8 km/h. Mo jogs at a constant speed of 6 km/h. Kari and Mo jog from the same starting point to the same finishing point along a straight road. Mo starts at 10:00 a.m. Kari and Mo both finish at 11:00 a.m. At what time did Kari start to jog?

    3. Full solutionThe line with equation \(x + 3y =7\) is parallel to the line with equation \(y = mx+b\). The line with equation \(y=mx+b\) passes through the point \((9,2)\). Determine the value of \(b\).

    1. LightbulbMichelle calculates the average of the following numbers: \[5, 10, 15, 16, 24, 28, 33, 37\] Daphne removes one number and calculates the average of the remaining numbers. The average that Daphne calculates is one less than the average that Michelle calculates. Which number does Daphne remove?

    2. LightbulbIf \(16^{^{\frac{\underset{15}{}}{\overset{}{x}}}}=32^{^{\frac{\underset{4}{}}{\overset{}{3}}}}\), what is the value of \(x\)?

    3. Full solutionSuppose that \(\dfrac{2^{2022}+2^{a}}{2^{2019}} = 72\). Determine the value of \(a\).

    1. LightbulbIn the diagram, \(\triangle ABC\) has a right angle at \(B\) and point \(D\) lies on \(AB\).

      If \(DB=10\), \(\angle ACD = 30^\circ\) and \(\angle CDB = 60^\circ\), what is the length of \(AD\)?

    2. Full solutionThe points \(A(d,-d)\) and \(B(-d+12, 2d-6)\) both lie on a circle centered at the origin. Determine the possible values of \(d\).

    1. LightbulbDetermine the two pairs of positive integers \((a,b)\) with \(a<b\) that satisfy the equation \(\sqrt{a}+\sqrt{b} = \sqrt{50}\).

    2. Full solutionConsider the system of equations: \[\begin{aligned} c + d & = 2000\\ \dfrac{c}{d} & = k\end{aligned}\] Determine the number of integers \(k\) with \(k \geq 0\) for which there is at least one pair of integers \((c,d)\) that is a solution to the system.

    1. LightbulbA regular pentagon covers part of another regular polygon, as shown.

      The regular polygon has a radius slightly smaller than the pentagon. The approximate positions of the overlapping shapes are given as follows.

      This regular polygon has \(n\) sides, five of which are completely or partially visible. In the diagram, the sum of the measures of the angles marked \(a^\circ\) and \(b^\circ\) is \(88^\circ\). Determine the value of \(n\).

      (The side lengths of a regular polygon are all equal, as are the measures of its interior angles.)

    2. Full solutionIn trapezoid \(ABCD\), \(BC\) is parallel to \(AD\) and \(BC\) is perpendicular to \(AB\).

      Place vertex B with the right angle at the bottom left. One adjoining vertex lies directly to the right of B. The other adjoining vertex lies directly above B.

      Also, the lengths of \(AD\), \(AB\) and \(BC\), in that order, form a geometric sequence. Prove that \(AC\) is perpendicular to \(BD\).

      (A geometric sequence is a sequence in which each term after the first is obtained from the previous term by multiplying it by a non-zero constant.)

    1. Full solutionDetermine all real numbers \(x\) for which \(2\log_2(x-1) = 1 - \log_2(x+2)\).

    2. Full solutionConsider the function \(f(x)=x^2-2x\). Determine all real numbers \(x\) that satisfy the equation \(f(f(f(x)))=3\).

    1. Full solutionA circle has centre \(O\) and radius 1. Quadrilateral \(ABCD\) has all 4 sides tangent to the circle at points \(P\), \(Q\), \(S\), and \(T\), as shown.

      P lies on AB, Q lies on BC, S lies on CD, and T lies on DA.

      Also, \(\angle AOB = \angle BOC = \angle COD = \angle DOA\). If \(AO=3\), determine the length of \(DS\).

    2. Full solutionSuppose that \(x\) satisfies \(0<x<\dfrac{\pi}{2}\) and \(\cos \left(\dfrac{3}{2} \cos x\right)=\sin \left(\dfrac{3}{2}\sin x\right)\). Determine all possible values of \(\sin 2x\), expressing your answers in the form \(\dfrac{a\pi^2 + b\pi + c}{d}\) where \(a,b,c,d\) are integers.

  9. Full solutionFor positive integers \(a\) and \(b\), define \(f(a,b) = \dfrac{a}{b} + \dfrac{b}{a} + \dfrac{1}{ab}\). For example, the value of \(f(1,2)\) is \(3\).

    1. Determine the value of \(f(2,5)\).

    2. Determine all positive integers \(a\) for which \(f(a,a)\) is an integer.

    3. If \(a\) and \(b\) are positive integers and \(f(a,b)\) is an integer, prove that \(f(a,b)\) must be a multiple of 3.

    4. Determine four pairs of positive integers \((a,b)\), with \(2<a<b\), for which \(f(a,b)\) is an integer.

  10. Full solution

    1. Amir and Brigitte play a card game. Amir starts with a hand of 6 cards: 2 red, 2 yellow and 2 green. Brigitte starts with a hand of 4 cards: 2 purple and 2 white. Amir plays first. Amir and Brigitte alternate turns. On each turn, the current player chooses one of their own cards at random and places it on the table. The cards remain on the table for the rest of the game. A player wins and the game ends when they have placed two cards of the same colour on the table. Determine the probability that Amir wins the game.

    2. Carlos has \(14\) coins, numbered \(1\) to \(14\). Each coin has exactly one face called “heads”. When flipped, coins \(1,2,3,\ldots,13,14\) land heads with probabilities \(h_1, h_2, h_3, \dots, h_{13}, h_{14}\), respectively. When Carlos flips each of the \(14\) coins exactly once, the probability that an even number of coins land heads is exactly \(\frac{1}{2}\). Must there be a \(k\) between 1 and 14, inclusive, for which \(h_k = \frac{1}{2}\)? Prove your answer.

    3. Serge and Lis each have a machine that prints a digit from 1 to 6. Serge’s machine prints the digits \(1,2,3,4,5,6\) with probability \(p_1\), \(p_2\), \(p_3\), \(p_4\), \(p_5\), \(p_6\), respectively. Lis’s machine prints the digits \(1,2,3,4,5,6\) with probability \(q_1\), \(q_2\), \(q_3\), \(q_4\), \(q_5\), \(q_6\), respectively. Each of the machines prints one digit. Let \(S(i)\) be the probability that the sum of the two digits printed is \(i\). If \(S(2)=S(12) = \frac{1}{2}S(7)\) and \(S(7)>0\), prove that \(p_1=p_6\) and \(q_1=q_6\).

Further Information

For students...

Thank you for writing the Euclid Contest!

If you are graduating from secondary school, good luck in your future endeavours! If you will be returning to secondary school next year, encourage your teacher to register you for the Canadian Senior Mathematics Contest, which will be written in November.

Visit our website cemc.uwaterloo.ca to find

For teachers...

Visit our website cemc.uwaterloo.ca to